Tag Archives: Geometry

The study of Plane Practical Geometry starts with “The Elements” of Euclid. Euclidean geometry is basically planar or flat plane geometry. We will first look at the foundation of Geometry. Geometric literature entails the following:

[A] Initial notions – There are some notions in geometry (and in mathematics in general), to which it is impossible to give some sensible definition. We adopt them as initial notions. The meaning of these notions can be ascertained only by experience. So, the notions of a point and a straight line are initial. Based on initial notions we can give definitions to all other notions.

[B] Axiom(s) – a statement, determining some property and adopted without a proof. Axioms have been arisen by experience and the experience checks if they are true in totality.

[C] Proof(s) – a reasoning, determining some property.

[D] Theorem(s) – a statement, determining some property and requiring a proof. Theorems are called also as lemmas, properties, consequences, rules, criteria, propositions, statements.

Coming back to our main topic of discussion here, you can very well see that at the onset, there are few “beliefs” that we take for granted and build the whole of geometry based on those set of beliefs. The initial notions are of a point and a straight line.

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Theory: A point is a dimensionless indivisible whole. It is an infinitesimal whole without a part. If a person were to reject such a notion at the very outset saying that it does not exist then the whole foundation of geometry and its applications would collapse.

Between any two points whether in neighbourhood or not, there are an infinite set of points in between that can be joined to form a line segment of finite length and can be extended on one or both sides to form either a ray or a line.

A line is a doubly directed continuous infinite length without a breadth. A ray is a singly directed continuous infinite length without a breadth.

Practicality threatened: A carpenter won’t be able to produce a single piece of furniture if he were to mock at the concept of a point (although it does not exist in reality). He just won’t be able to do any valid markings nor take any measurements because if he does not believe in “the point”, how can he believe in the line or a line segment? Lines and line segments are produced by points. He cannot go further if he disbelieves in the concept of a point. Also, practically lines cannot exist without breadth.

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Theory: Euclid further states that if there are 3 points OR a line and a point outside the line, then they describe one plane.

Practical Aspect: In carpentry, it is a common practice to construct a tripod or a three-legged stool for greater stability than a four-legged one.

The reason for this being that the 4-legged stool can topple over a surface which is not planar or flat, but a 3-legged stool will never topple over an uneven, non-flat surface. The 3 legs always lie in one plane or the other which imparts them stability and prevents wobbling.

Reconciling Theory and Practicality: Here in this case, the point which was infinitesimal is considered to have the footprint of the leg of a stool. The concept comes handy. We can say that the footprint of the point takes upon a value which is finite in area and itself contains infinite points in the footprint.

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We now come close to the idea of “Zooming in” and “Zooming out” of points.

Explanation: We see the moon in the sky bigger than the sun. It is only because of their relative distances from the earth that it appears so. It is factual data that if we travel away from our own solar system, then the planets and our sun itself will start appearing tiny and become more like dots in space from a sufficient distance.

Euclid’s theory does not contain “Zoom in” and “Zoom out” features on points and lines. From a practical point of view however, these features can be readily put to use even in Satellite communication, navigation and imaging.

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Theory: Two lines on the same plane are either parallel or intersecting.

Practical point of view: Euclid had assumed a flat plane. Consider a plane which is spherical. Let us consider a small replica of the earth which is actually oblong as spherical for purposes of simplicity. We can easily draw an equator and then we can draw a circle on the upper hemisphere which is neither parallel nor intersecting the equator.

For easy visualization, consider the upper hemisphere sliced off from the lower. Now consider only the upper hemisphere and then slice off a part of the upper hemisphere at an acute angle transversely, not touching the equatorial plane.

Now join the sections back to form the “replica earth”. The lines where we have cut appear to be neither parallel nor intersecting. They are anti-parallel circles. The sectional plane and equatorial plane would intersect only outside the sphere.

Alternately, two lines which are neither parallel nor intersecting may be skew lines on two parallel planes. E.g. Anti-parallel diagonals of a cube belonging to its lower and upper faces.

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Theory: Two lines perpendicular to the same line are parallel.

Practical point of view: The above is true only for a flat plane. Draw two lines perpendicular to the equator on a spherical plane and they would intersect at the north and south poles……(Ref.-1)

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Theory: The sum of the interior angles of a triangle equals 180 degrees.

Practical point of view: The above is true only for a flat plane. Consider the triangle described (in Ref.-1) above. It is possible to have a triangle with all 3 interior angles to be equal to 90 degrees OR 2 internal right angles and 1 acute angle OR 2 internal right angles and 1 obtuse angle barring the exception of a single diametric vertical section in which a triangle won’t be formed at all.

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Conclusion: The geometric truths known to us are fairly relative both in theory and application. Generalizations are difficult to make based on deferring truths.

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God:

Theories are formed out of the combination of some inner and outer necessities but they have to be viewed in appropriate ways for correct applications to be well grounded in practicality. There are and will be stark differences between theories and practicalities but they are illuminating indeed for the mind that ponders upon and realizes them and needs to aptly apply them. As we saw in geometry, now I am going to elucidate the geometry in the Creation of God, alias of course “The Universe”.

The Universe is born out of an ellipsoid (rotate an ellipse 360 degrees around its own major axis and the resulting 3D form is an ellipsoid) which was golden yellow in colour. It was called “Hiranyagarbha” or the Golden egg OR the Golden embryo OR the Golden foetus in classical literature such as the Vedas and Puranas. In fact, the Sanskrit word for the universe is “Brahmanda” which itself means Cosmic Egg. Another diadem is the mention of “Lingaakara Srishti”, which means Creation in the form of multiples of Lingas or Ellipsoids – one resulting from the other.

The chain was:- From Brahmanda-Linga was created Atma-Linga. From Atma-Linga was created Gyana-Linga from which was created Prana-Linga. The Brahmanda-Linga itself evolved out of Paraatpara Shiv-Linga OR Sadashiva-Linga in which was represented the union of Male and Female Templates in their purest form. The Male template was a Linga and the Female template was a Yoni in the highest form. (The Yoni resembles a doughnut in shape). The friction in these two templates — the Linga and Yoni which got aligned to each other created 2 patterns of consciousness – one upper and one lower (like its reflection in water).

These two patterns in total represented 14 layers of consciousness – 7 each from each pattern. The pattern was of the Sri Chakra and was multidimensional in nature. One Sri Chakra which was above, started in its uppermost tip which was Nirguna Paramatma or Pure God Consciousness. Similarly, the inverted Sri Chakra which was below culminated in its lowermost tip which was Avaguna Anaatma or Impure Devil Consciousness. The Earth lies in the Lowest Layer of the Upper Sri Chakra. Thus it is approximately in the middle of creation albeit a little over it.

Without loss of integrity, instead of digressing or divulging from the main point of focus, let us get back to the geometry of our own self. We have five bodies out of which 2nd (the etheric body or Pranamaya kosha) and 4th body (the causal body or the Gyanamaya kosha) along with the soul are Lingaakara or Ellipsoids. The 1st body in the order of visibility from gross to subtle (the physical body) and the 3rd body (the mind body or the astral body) act like yonis. The 5th body (the bliss body or Anandamaya kosha) is a perfect yoni. Atma-Linga interacts with the 5th body and forms Gyana-Linga (4th body) out of the union. The Gyana-Linga merges with the 3rd body and forms Prana-Linga (2nd body). The Prana-Linga conjugates with the physical body or the gross body (1st body) and a visible life form evolves. A personality is formed.

But this was creation that took place directly from God. Since the first incarnation of the soul on earth, only the Prana-Linga conjugated with the physical body. From the influence of outside environment and wrong or improper interaction between Prana-Linga and the physical body, the physical body got bad unworkable inputs which it carried to the higher bodies as contaminations—one up at a time.

Ultimately, they got accumulated as negative samskaras or negative psychic impressions in the Gyana-Linga. The Gyana-Linga thus got distorted and lost its even shape. It lost its lustre too. It started to gather heaps and heaps of samskaras – both good and bad according to the karma or deeds of the person. The Dhyana Bindu Upanishad states that there are thousands of mountains of negative psychic impressions stored in the 4th body – the Gyana-Linga of a human being. Instead of clearing the debris, a human being starts gathering more psychic impressions in each successive janma or re-incarnation adding to the garbage of mountains of black energy.

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Summary of this lengthy discussion:

We think in fairly limited ways like in Euclid’s geometry and try to engulf in 3D exchanges of gross matter with monetary affairs. Our ignorance is far greater than knowledge. One change in axiomatic and proof-oriented thinking can shake our very foundations of existence. Stop behaving like automatons. Use your power of constructive thinking and do your bit in orienting planet earth towards more positivism.

Today’s fast-paced age demands information but that too has been centred around materialism and consumerism. Creation itself being multidimensional, it is high time that a human being should think plausibly for himself in holistic ways that far exceed today’s material science and religion-centred spirituality.